Chapter 9: Sets and Maps 1

• The Map and Set interfaces and how to use them
• Hash coding and its use in efficient search & retrieval
• Two forms of hash tables:
• Open addressing
• Chaining
• Their relative benefits and performance tradeoffs
• Implementing both hash table forms
• Introduction to implementation of Maps and Sets
• Applying Maps and Sets to previous problems
Chapter 9: Sets and Maps 2

• This part of the Collection hierarchy includes 3 interfaces, 2 abstract classes, and 2 actual classes
Chapter 9: Sets and Maps 3

• A set is a collection containing no duplicate elements
• Operations on sets include:
• Testing for membership
• Adding elements
• Removing elements
• Union
• Intersection
• Difference
• Subset
Chapter 9: Sets and Maps 4

// element oriented methods boolean contains (E e) // member test boolean add (E e) // enforces no-dups boolean remove (Object o) boolean isEmpty () int size ()
Iterator<E> iterator ()
Chapter 9: Sets and Maps 5

// Set/Collection oriented methods boolean containsAll (Collection<E> c)
// subset test boolean addAll (Collection<E> c)
// set union boolean removeAll (Collection<E> c)
// set difference boolean retainAll (Collection<E> c)
// set intersection
Chapter 9: Sets and Maps 6

• Constructors enforce the “no duplicates” criterion
• Add methods do not allow duplicates either
• Certain methods are optional
• add, addAll, remove, removeAll, retainAll
Chapter 9: Sets and Maps 7

String[] aA = {“Ann”,”Sal”,”Jill”,”Sal”};
String[] aB = {“Bob”,”Bill”,”Ann”,”Jill”};
Set<String> sA = new HashSet<String>();
// HashSet implements Set
Set<String> sA2 = new HashSet<String>();
Set<String> sB = new HashSet<String>(); for (String s : aA) { sA.add(s); sA2.add(s);
} for (String s : aB) { sB.add(s);
}
Chapter 9: Sets and Maps 8

...
System.out.println(“The two sets are:\n” + sA + “\n” + sB); sA.addAll(sB); // union sA2.retainAll(sB); // intersection
System.out.println(“Union: “, sA);
System.out.println(“Intersection: “, sA2);
Chapter 9: Sets and Maps 9

• Sets allow no duplicates
• Sets do not have positions , so no get or set method
• Set iterator can produce elements in any order
Chapter 9: Sets and Maps 10

Map
• Map is related to Set: it is a set of ordered pairs
• Ordered pair: ( key , value )
• In a given Map , there are no duplicate keys
• Values may appear more than once
• Can think of key as “mapping to” a particular value
• Maps support efficient organization of information in tables
• Mathematically, these maps are:
• Many-to-one (not necessarily one-to-one)
• Onto (every value in the map has a key)
Chapter 9: Sets and Maps 11

Chapter 9: Sets and Maps 12

Map
// some methods of java.util.Map<K, V>
// K is the key type
// V is the value type
V get (Object key)
// may return null
V put (K key, V value)
// returns previous value or null
V remove (Object key)
// returns previous value or null boolean isEmpty () int size ()
Chapter 9: Sets and Maps 13

// this builds Map in previous picture
Map<String, String> m = new HashMap<String, String>();
// HashMap is an implementation of Map m.put(“J” , “Jane”); m.put(“B” , “Bill”); m.put(“S” , “Sam” ); m.put(“B1”, “Bob” ); m.put(“B2”, “Bill”);
//
System.out.println(“B1->“ + m.get(“B1”));
System.out.println(“Bill->”+m.get(“Bill));
Chapter 9: Sets and Maps 14

// Idea: enter word once
// with list of lines on which it occurs
... inner loop: word has the word ...
// get list of lines for this word
ArrayList<Integer> lines = index.get(word); if (lines == null) { lines = new ArrayList<Integer>(); index.put(word, lines);
} lines.add(lineNum);
...
Chapter 9: Sets and Maps 15

• Goal: access item given its key (not its position )
• Therefore, want to locate it directly from the key
• In other words, we wish to avoid much searching
• Hash tables provide this capability
• Constant time in the average case!
• Linear time in the worst case
O(1)
O(n)
• Searching an array: O(n) Searching BST: O(log n)
Chapter 9: Sets and Maps 16

• Suppose we have a table of size N
• A hash code is:
• A number in the range 0 to N-1
• We compute the hash code from the key
• You can think of this as a “default position” when inserting, or a “position hint” when looking up
• A hash function is a way of computing a hash code
• Desire: The set of keys should spread evenly over the
N values
• When two keys have the same hash code: collision
Chapter 9: Sets and Maps 17

• Want to count occurrences of each Character in a file
• There are 2 16 possible characters, but ...
• Maybe only 100 or so occur in a given file
• Approach: hash character to range 0-199
• That is, use a hash table of size 200
• A possible hash function for this example: int hash = unicodeChar % 200;
• Collisions are certainly possible (see later)
Chapter 9: Sets and Maps 18

• Simple functions often produce many collisions
• ... but complex functions may not be good either!
• It is often an empirical process
• Adding letter values in a string: same hash for strings with same letters in different order
• Better approach: int hash = 0; for (int i = 0; i < s.length(); ++i) hash = hash * 31 + s.charAt(i);
• This is the hash function used for String in Java
Chapter 9: Sets and Maps 19

• The String hash is good in that:
• Every letter affects the value
• The order of the letters affects the value
• The values tend to be spread well over the integers
• Table size should not be a multiple of 31:
• Calculate index: int index = hash % size;
• For short strings, index depends heavily on the last one or two characters of the string
• They chose 31 because it is prime, and this is less likely to happen
Chapter 9: Sets and Maps 20

• Guidelines for good hash functions:
• Spread values evenly : as if “random”
• Cheap to compute
• Generally, number of possible values >> table size
Chapter 9: Sets and Maps 21

• Will consider two ways to organize hash tables
• Open addressing
• Chaining
• Open addressing:
• Hashed items are in a single array
• Hash code gives position “hint”
• Handle collisions by checking multiple positions
• Each check is called a probe of the table
Chapter 9: Sets and Maps 22

• Probe by incrementing the index
• If “fall off end”, wrap around to the beginning
• Take care not to cycle forever!
1. Compute index as hashCode() % table.length
2. if table[index] == null, item is not in the table
3. if table[index] matches item, found item (done)
4. Increment index circularly and go to 2
• Why must we probe repeatedly?
• hashCode may produce collisions
• remainder by table.length may produce collisions
Chapter 9: Sets and Maps 23

Ways to obtain proper termination
• Stop when you come back to your starting point
• Stop after probing N slots, where N is table size
• Stop when you reach the bottom the second time
• Ensure table never full
• Reallocate when occupancy exceeds threshold
Chapter 9: Sets and Maps 24

• Cannot traverse a hash table
• Order of stored values is arbitrary
• Can use an iterator to produce in arbitrary order
• When item is deleted, cannot just set its entry to null
• Doing so would break probing
• Must store a “dummy value” instead
• Deleted items waste space and reduce efficiency
• Use prime number for table size: reduces collisions
• Higher occupancy causes makes for collisions
Chapter 9: Sets and Maps 25

• Table of strings, initial size 5
• Add “Tom”, hash 84274  4
• Add “Dick”, hash 2129869  4
• Add “Harry”, hash 69496448  3
• Add “Sam”, hash 82879  4
• Add “Pete”, hash 2484038  3
Slot 4
Slot 0 (wraps)
Slot 3
Slot 1 (wraps)
Slot 2 (wraps)
• Note: many lookups will probe a lot!
• Size 11 gives these slots: 3, 5, 10, 5  6, 7
Chapter 9: Sets and Maps 26

• Choose a new larger size, e.g., doubling
• (Re)insert non-deleted items into new array
• Install the new array and drop the old
• Similar to reallocating an ArrayList, etc.
• But , elements can move around in reinsertion
• Hope: rehashing distributes items at least as well
Chapter 9: Sets and Maps 27

• Linear probing
• Tends to form long clusters of keys in the table
• This causes longer search chains
• Quadratic probing can reduce the effect of clustering
• Index increments form a quadratic series
• Direct calculation involves multiply, add, remainder
• Incremental calculation better (in a moment)
• Probe sequence may not produce all table slots
Chapter 9: Sets and Maps 28

• Generating the quadratic sequence
Want: s, s+1 2 , s+2 2 , s+3 2 , s+4 2 , etc. (all % length)
“Trick” to calculate incrementally:
Initially: int index = ... 1 st probe slot ...
int k = -1;
At each iteration: k += 2; index = (index + k) % table.length;
Chapter 9: Sets and Maps 29

• Alternative to open addressing
• Each table slot references a linked list
• List contains all items that hash to that slot
• The linked list is often called a bucket
• So sometimes called bucket hashing
• Examines only items with same hash code
• Insertion about as complex
• Deletion is simpler
• Linked list can become long  rehash
Chapter 9: Sets and Maps 30

Two items hashed to bucket 3
Three items hashed to bucket 4
Chapter 9: Sets and Maps 31

• Load factor = # filled cells / table size
• Between 0 and 1
• Load factor has greatest effect on performance
• Lower load factor  better performance
• Reduce collisions in sparsely populated tables
• Knuth gives expected # probes p for open addressing, linear probing, load factor L: p = ½(1 + 1/(1-L))
• As L approaches 1, this zooms up
• For chaining, p = 1 + (L/2)
• Note: Here L can be greater than 1!
Chapter 9: Sets and Maps 32

Chapter 9: Sets and Maps 33

• Hash table:
• Insert: average O(1)
• Search: average O(1)
• Sorted array:
• Insert: average O(n)
• Search: average O(log n)
• Binary Search Tree:
• Insert: average O(log n)
• Search: average O(log n)
• But balanced trees can guarantee O(log n)
Chapter 9: Sets and Maps 34

• Hash table:
• Open addressing space: n/L e.g., 1.5 to 2 x n
• Chaining: assuming 4 words per list node (2 header, 1 next, 1 data): n(1+4L)
• Sorted array:
• Space: n
• Binary Search Tree:
• Space: 5n (5 words per tree node: 2 header, 1 left,
1 right, 1 data)
Chapter 9: Sets and Maps 35

• Interface HashMap : used for both implementations
• Class Entry : simple class for (key, value) pairs
• Class HTOpen : implements open addressing
• Class HTChain : implements chaining
• Further implementation concerns
Chapter 9: Sets and Maps 36

HashMap<K,V>
Note: Java API version has many more operations!
V get (Object key)
// may return null
V put (K key, V value)
// returns previous value; null if none
V remove (Object key)
// returns previous value; null if none boolean isEmpty () int size ()
Chapter 9: Sets and Maps 37

Entry private static class Entry<K, V> { private K key; private V value; public Entry (K key, V value) { this.key = key; this.value = value;
} public K getKey () { return key; } public V getValue () { return value; } public V setValue (V newVal) {
V oldVal = value; value = newVal; return oldVal;
} }
Chapter 9: Sets and Maps 38

HTOpen<K,V> public class HTOpen<K, V> implements HashMap<K, V> { private Entry<K, V>[] table; private static final int INIT_CAP = 101; private double LOAD_THRESHOLD = 0.75; private int numKeys; private int numDeletes;
// special “marker” Entry private final Entry<K, V> DELETED = new Entry<K, V>(null, null); public HTOpen () { table = new Entry[INIT_CAP];
}
... // inner class Entry can go here
Chapter 9: Sets and Maps 39

HTOpen<K,V>
private int find (Object key) { int hash = key.hashCode(); int idx = hash % table.length; if (idx < 0) idx += table.length; while ((table[idx] != null) &&
(!key.equals(table[idx].key))) { idx++; if (idx >= table.length) idx = 0;
// could do above 3 lines as:
// idx = (idx + 1) % table.length;
} return idx;
}
Chapter 9: Sets and Maps 40

HTOpen<K,V>
public V get (Object key) { int idx = find(key); if (table[idx] != null) return table[idx].value; else return null;
}
Chapter 9: Sets and Maps 41

HTOpen<K,V>
public V put (K key, V val) { int idx = find(key); if (table[idx] == null) { table[idx] = new Entry<K,V>(key,val); numKeys++; double ldFact = // NOT int divide!
(double)(numKeys+numDeletes) / table.length; if (ldFact > LOAD_THRESHOLD) rehash(); return null;
}
V oldVal = table[idx].value; table[idx].value = val; return oldVal; }
Chapter 9: Sets and Maps 42

HTOpen<K,V>
private void rehash () {
Entry<K, V>[] oldTab = table; table = new Entry[2*oldTab.length + 1];
// the + 1 keeps length odd numKeys = 0; numDeletes = 0; for (int i = 0; i < oldTab.length; ++i){ if ((oldTab[i] != null) &&
(oldTab[i] != DELETED)) { put(OldTab[i].key, oldTab[i].value);
}
}
}
// The remove operation is an exercise
Chapter 9: Sets and Maps 43

HTChain<K,V> public class HTChain<K, V> implements HashMap<K, V> { private LinkedList<Entry<K, V>>[] table; private int numKeys; private static final int CAPACITY = 101; private static final double
LOAD_THRESHOLD = 3.0;
// put inner class Entry here public HTChain () { table = new LinkedList[CAPACITY];
}
...
}
Chapter 9: Sets and Maps 44

HTChain<K,V>
public V get (Object key) { int hash = key.hashCode(); int idx = hash % table.length; if (idx < 0) idx += table.length; if (table[idx] == null) return null; for (Entry<K, V> item : table[idx]) { if (item.key.equals(key)) return item.value;
} return null;
}
Chapter 9: Sets and Maps 45

HTChain<K,V>
public V put (K key, V val) { int hash = key.hashCode(); int idx = hash % table.length; if (idx < 0) idx += table.length; if (table[idx] == null) table[idx] = new LinkedList<Entry<K, V>>(); for (Entry<K, V> item : table[idx]) { if (item.key.equals(key)) {
V oldVal = item.value; item.value = val; return oldVal;
}
} // more ....
Chapter 9: Sets and Maps 46

HTChain<K,V>
// rest of put: “not found” case table[idx].addFirst( new Entry<K, V>(key, val)); numKeys++; if (numKeys >
(LOAD_THRESHOLD * table.length)) rehash(); return null;
}
// remove and rehash left as exercises
Chapter 9: Sets and Maps 47

• Class Object implements hashCode and equals
• Every class has these methods
• One may override them when it makes sense to
• Object.equals
compares addresses, not contents
• Object.hashCode
based on address, not contents
• Java recommendation:
• If you override equals , then
• you should also override hashCode
Chapter 9: Sets and Maps 48

equals
hashCode
• Consider class Person with field IDNum public boolean equals (Object o) { if (!(o instanceof Person)) return false; return IDNum.equals(((Person)o).IDNum);
}
• Demands a matching hashCode method: public int hashCode () {
// equal objects will have equal hashes return IDNum.hashCode();
}
Chapter 9: Sets and Maps 49

• Can use HashMap<E,E> and pairs (key,key)
• This is an adapter class
• Can use an Entry<E> inner class
• Can implement with an E array
• In each case, can code open addressing and chaining
• The coding of each method is analogous to what we saw with HashMap
Chapter 9: Sets and Maps 50

• The Java API uses a hash table to implement both the
Map and Set interfaces
• Implementing them is aided by abstract classes
AbstractMap and AbstractSet in the
Collection hierarchy
• Interface Map requires nested type Map.Entry<K,V>
• Interface Map also requires support for viewing it as a
Set of Entry objects
Chapter 9: Sets and Maps 51

public String addOrChangeEntry (
String name, String newNum) {
String oldNum = dir.put(name, newNum); modified = true; return oldNum;
} public String lookupEntry (String name) { return dir.get(name);
} public String removeEntry (String name) {
String ret = dir.remove(name); if (ret != null) modified = true; return ret; }
Chapter 9: Sets and Maps 52

// in loadData: while ((name = ins.readLine()) != null) { if ((number = ins.readLine()) == null) break; dir.put(name, number);
}
// saving for (Map.Entry<String,String> curr : dir.entrySet()) { outs.println(curr.getKey()); outs.println(curr.getValue());
}
Chapter 9: Sets and Maps 53

// First, want to build frequency table
// for a given input file public static HuffData[] buildFreqTable (
BufferedRead ins) {
Map<Character, Integer> freqs = new HashMap<Character, Integer>(); try {
... process each character ...
} catch (IOException ex) { ex.printStackTrace();
}
... build array from map ...
Chapter 9: Sets and Maps 54

// process each character int next; while ((next = ins.read()) != -1) {
Integer count = freqs.get((char) next); if (count == null) count = 1; else
++count; freqs.put((char)next, count);
} ins.close();
Chapter 9: Sets and Maps 55

// build array from map
HuffData[] freqTab = new HuffData[freqs.size()]; int i = 0; for (Map.Entry<Character,Integer> entry : freqs.entrySet()) { freqTab[i++] = new HuffData( entry.getValue().doubleValue(), entry.getKey());
} return freqTab;
Chapter 9: Sets and Maps 56

// build ENCODING table public void buildCodeTab () { codeMap = new HashMap<Character,BitString>(); buildCodeTab(huffTree, new BitString());
}
Chapter 9: Sets and Maps 57

public void buildCodeTab (
BinaryTree<HuffData> tree,
BitString code) {
HuffData datum = tree.getData(); if (datum.symbol != null) codeMap.put(datum.symbol, code); else {
BitString l = (BitString)code.clone(); l.append(false); buildCodeTab(tree.left() , l);
BitString r = (BitString)code.clone(); r.append(true); buildCodeTab(tree.right(), r); } }
Chapter 9: Sets and Maps 58

public void encode (BufferedReader ins,
ObjectObjectStream outs) {
BitString res = new BitString(); try { int next; while ((next = ins.read()) != -1) {
Character nxt = (char)next;
BitString nextChunk = codeMap.get(nxt); res.append(nextChunk);
} res.trimCapacity(); ins.close(); outs.writeObject(res);outs.close();...
Chapter 9: Sets and Maps 59